1. The problem is to understand and use partial derivatives. 2. Partial derivatives measure how a multivariable function changes as one variable changes, keeping others constant. 3. For a function $f(x,y)$, the partial derivative with respect to $x$ is denoted $\frac{\partial f}{\partial x}$ and with respect to $y$ is $\frac{\partial f}{\partial y}$. 4. To compute $\frac{\partial f}{\partial x}$, treat $y$ as a constant and differentiate $f$ with respect to $x$. 5. Similarly, to compute $\frac{\partial f}{\partial y}$, treat $x$ as a constant and differentiate $f$ with respect to $y$. 6. Example: Given $f(x,y) = x^2 y + 3xy^2$, find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$. 7. Compute $\frac{\partial f}{\partial x}$: $$\frac{\partial}{\partial x}(x^2 y + 3xy^2) = 2x y + 3 y^2$$ 8. Compute $\frac{\partial f}{\partial y}$: $$\frac{\partial}{\partial y}(x^2 y + 3xy^2) = x^2 + 6 x y$$ 9. These partial derivatives tell us how $f$ changes as $x$ or $y$ changes independently. Final answer: $$\frac{\partial f}{\partial x} = 2 x y + 3 y^2$$ $$\frac{\partial f}{\partial y} = x^2 + 6 x y$$